Contaminant mass discharge across a control plane downstream of a Dense Nonaqueous Phase Liquid (DNAPL) source zone has great potential to serve as a metric for the assessment of the effectiveness of source zone treatment technologies and for the development of risk-based source-plume remediation strategies. However, field estimated mass discharge is always subject to great uncertainty arising from non-exclusive sampling. The accuracy of the mass discharge estimate and the magnitude of its quantifiable uncertainty depend upon the amount of information provided by the sample data.
The difficulties encountered in the remediation of DNAPL in the subsurface have instigated numerous researches in the benefits evaluation of DNAPL source zone depletion. Mass discharge, defined as the contaminant mass per unit time ([M/T]) migrating across a hypothetic control plane orthogonal to the mean groundwater flow, has been widely proposed as a potential metric to assess the impact of partial mass removal in DNAPL source zones (Interstate Technology & Regulatory Council, 2004, Strategies for Monitoring the Performance of DNAPL Source Zone Remedies; Soga et al., 2004, J. Haz. Mat. 110:13-26; Rao et al., 2002, Groundwater Quality 2001 Conference, Sheffield, UK, 275:571-578; Freeze and McWhorter, 1997, Ground Water 35:111-123; Feenstra et al., 1996, in Dense Chlorinated Solvents and Other DNAPLs in Groundwater, Waterloo Press, USA).
In the field, mass discharge prediction made from relative observations such as contaminant concentration, hydraulic conductivity, or mass flux ([M/T·L2]) is always subject to uncertainty (Jarsjo et al., 2005, J. Cont. Hydro. 79:107-134; Zeru and Schafer, 2005, J. Cont. Hyro. 81:106-124; Hatfield et al., 2004, J. Cont. Hydro. 75:155-181), however, the uncertainty is typically not quantified. This circumstance makes the implementation of mass discharge as an assessment metric extremely difficult (Stroo et al., 2003, Env. Sci. Tech. 37:224A-230A) since any decision regarding site management or benefits/cost evaluation using incomplete information is unreliable (Abriola, 2005, Env. Health Pers. 113:A438-A439). A geostatistical approach has been proposed to quantify the uncertainty of mass discharge using multi-level measurements of contaminant concentration and hydraulic conductivity. The fully characterized uncertainty, in terms of the probability distribution of mass discharge, is ready to serve risk-based source-plume remediation strategies. Unfortunately, implementation of this geostatistical approach on numerically simulated control planes (Christ et al., 2005, Water Resour. Res. 41:W01007; Christ et al., 2006, Water Resourc. Res. in press; Lemke et al., 2004, Water Resourc. Res. 40:W01511) suggests that the sampling density, defined as the proportion of the sampled area to the whole control plane, could be as high as 6˜7% to achieve an accurate model of uncertainty for the control planes with scattered small hot spots and large areas of near-zero concentration. This conclusion was made for the one stage sampling design with a regular sampling pattern (rectangular). The requirement of such a high sampling density is mainly due to the limitation of the geostatistical approach on non-representative samples.
Classical sampling techniques (Cochran, 1977, Sampling Techniques, John Wiley & Sons, Inc., New York, N.Y., pp. 428) typically offer limited help for sampling design when spatially distributed processes are encountered. Instead, geostatistics is usually employed for spatial sampling design (Thompson, 2002, Sampling, 2nd Ed. John Wiley & Sones, Inc., New York, N.Y., pp. 367; Isaaks and Srivastava, 1989, Applied Geostatistics, Oxford Univ. Press, New Tork, N.Y., pp. 561; Goovaerts and Journel, 1997, Geostatistics for Natural Resource Evaluation, Oxford Univ. Press, New York, N.Y.). The term spatial sampling design refers to the procedure that decides the arrangement of observations (e.g. sampling density and sampling locations) for specific purposes (Christakos and Olea, 1992, Adv. Water Res. 15:219-237). Different purposes result in different selection/formulation of sampling criteria, and thus lead to different sampling design.
One of the popular sampling criteria is the minimization of kriging variance (variance of kriging error), because kriging variance is independent of data values, only depending on the data configuration and the covariance structure. McBratney et al. (1981, Comp. & Geosci. 7:331-334) used this criterion to optimize the spacing of a sampling grid, given an a priori semivariogram exists. van Groenigen (2000, Geoderma 97:223-236) further discussed the impacts of the optimization criteria (minimization of mean kriging variance versus maximum kriging variance), semivariogram model type, and semivariogram model parameters. This criterion directly deals with the kriging error, which is closely corrected to the quality of the prediction of geostatistical method (kriging). The advantage of this method is that kriging variance by definition is independent of data values, so it is possible to be computed in advance of real sampling. However, the independency of data values also adversely impacts this criterion. For example, two locations with different contrast in the surrounding data values could have the same kriging variance, given the same data configuration (Goovaerts and Journel, 1997). Also, sampling cannot be preferentially guided in the region with high/low values.
This criterion is also very sensitive at the boundary of the study domain. The covariance matrix (or the semivariogram model) has to been known a priori, which is another disadvantage. Another type of criteria indirectly improves the results of geostatistical prediction by improving the quality of the information contained in sample data. For example, with no information to infer a reliable semivariogram in advance, van Groenigen and Stein (1998, J. Env. Quality 27:1078-1086) proposed to incorporate two sampling criteria: (1) the Minimization of the Mean of Shortest Distances Criterion (MMSD), and (2) The WM criterion, a criterion proposed by Warrick and Myers (1987, Water. Resourc. Res. 23:496-500). The MMSD criterion is for the coverage of the entire study area, which requires all observations spread evenly over the study area. The WM criterion is for the inference of a reliable semivariogram, which optimizes the distribution of observation pairs over different lag distances to an ideal distribution for the computation of an experimental semivariogram. The WM criterion is a complement to the MMSD criterion (but conflicting), because a sampling design with only MMSD will likely cause very few observation pairs for short lag distances and thus yield poor experimental semivariograms, especially with a limited number of observations.
Another example of this type of criterion is the study of Yfantis et al. (1987, Math. Geo. 19:183-205), which compared the performance of different sampling patterns on the computation of experimental semivariograms. By managing the quality of the information transmitted by observations for geostatistics, the ultimate purpose of this type of criteria is still to obtain reliable predictions from geostatistical techniques. The various natures of spatial surveys could yield other sampling purposes, thus different sampling criteria (Watson and Barnes, 1995, Math. Geo. 27:589-608; Christakos and Killam, 1993, Water Resourc. Res. 29:4063-4076; (Bogardi et al., 1985; Christakos and Olea, 1988, Adv. Water Res. 15:219-237; Ko et al., 1995, Oper. Res. 43:684-691; Bueso et al., 1998, Env. Eco. Stat. 5:29-44).
Once sampling criteria are appropriately selected, the remaining problem is the search for the optimal sampling locations. Most researchers treat this problem as an optimization problem. Sampling criteria are formulated in the objective function (fitness function), which is usually a function of coordinates (van Groenigen and Stein, 1998), or a function of grid spacing (thus sampling density) (Bogardi et al., 1985, Water Resourc. Res. 21:199-208). Various optimization algorithms have been used for the search, such as the simulated annealing algorithm (Dougherty and Marryott, 1991, Water Resourc. Res. 27:2493-2508; Christakos and Killam, 1993; van Groenigen et al., 1999, Geoderma 87:239-259) and the genetic algorithm (Cieniawski et al., 1995, Water Resourc. Res. 31:399-409; Savic and Walters, 1997, J. Water Resourc. Plan. Manag.—ASCE 123:67-77). However, two difficulties remained in this type of formulation. First, the quantities in the objective function should be able to be measured or computed before real sampling occurs, which limits the selection of sampling criteria. Second, the sampling density or sampling pattern should be known in advance. In case that sampling density is known, then the sampling locations are optimized. If sampling pattern has been decided, the optimization is for the sampling density (grid spacing). Moreover, it is common to include multiple sampling criteria and the objective function is usually the weighted average of these criteria. However, the weights are usually decided arbitrarily in advance. For example, equal weights are commonly used after each item in the objective function and are appropriately scaled (Nunes et al., 2004, J. Water Resourc. Plan. Manag.—ASCE 130:33-43). More advanced combinations of different weights are used to discuss the trade-off of each criterion (Bogardi et al., 1985; Meyer et al., 1994, Water Resourc. Res. 30:2647-2659; Cieniawski et al., 1995).
Another method is to implicitly perform a multi-stage sampling, where each stage has different pre-set weights and the sampling stages are decided according to a pre-set “cooling” schedule (Shieh et al., 2005, Math. Geo. 37:29-48).
In view of the above, what is needed is a sampling design that enables the selection of optimal sampling locations and that determines the minimal sampling density for an accurate quantification of mass discharge uncertainty in a field.